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Strongly interacting spin-orbit coupled Bose-Einstein condensates in one dimension

We theoretically study dilute superfluidity of spin-1 bosons with antiferromagnetic interactions and synthetic spin-orbit coupling (SOC) in a one-dimensional lattice. Employing a combination of density matrix renormalization group and quantum field theoretical techniques we demonstrate the appearance of a robust superfluid spin-liquid phase in which the spin-sector of this spinor Bose-Einstein condensate remains quantum disordered even after introducing quadratic Zeeman and helical magnetic fields. Despite remaining disordered, the presence of these symmetry breaking fields lifts the perfect spin-charge separation and thus the nematic correlators obey power-law behavior. We demonstrate that, at strong coupling, the SOC induces a charge density wave state that is not accessible in the presence of linear and quadratic Zeeman fields alone. In addition, the SOC induces oscillations in the spin and nematic expectation values as well as the bosonic Green's function. These non-trivial effects of a SOC are suppressed under the application of a large quadratic Zeeman field. We discuss how our results could be observed in experiments on ultracold gases of $^{23}$Na in an optical lattice.